5x²  +  17x  +31

         x²  -3x + 4 /^{----------------------------------------------------}

                          5x⁴     +2x³   +0x²    -9x    +12

                          5x⁴    -15x³ +20x²

                                   17x³   -20x²

                                   17x³  -51x²  +68x  

                                            31x²   -77x

                                            31x²  -93x  +124

                                                      16x - 112

So the polynomial and modulus:

5x² +17x +31 + (16x - 112)/(x² -3x +4)                                                                                  

                                       5x² + 17x + 31

x² - 3x + 4  |  5x⁴ + 2x³ +  0x²  -  9x +  12

                    5x⁴ - 15x³ + 20x²

                            17x³ - 20x²  -  9x

                            17x³ - 51x² + 68x

                                     31x²  -  77x +  12

                                     31x²  -  93x + 124

                                                16x  -  112

Thus 5x⁴ + 2x³ - 9x + 12 = (5x² + 17x + 31)(x² - 3x + 4) + 16x - 112